Tag: real numbers on number line

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Multiple choice maths real number fundemental theorem of arithmetic real numbers on number line fundamental theorem of arithmetic

Say true or false:
A positive integer is of the form $3q + 1,$ $q$  being a natural number, then you write its square in any form other than  $3m + 1$, i.e.,$ 3m $ or $3m + 2$  for some integer $m$.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the positive integer $n$ is of the form $3q, 3q+1,$ and $ 3q+2$
If $n=3q$
Squaring both sides, we get,
    $=>{ n }^{ 2 }=9{ q }^{ 2 }$
    $=>{ n }^{ 2 }=3\left( { 3q }^{ 2 } \right) $
    $=>{ n }^{ 2 }=3m$, where $m=3{ q }^{ 2 }$
Now, if $n=3q+1$
    $=>{ n }^{ 2 }={ \left( 3q+1 \right)  }^{ 2 }$
    $=>{ n }^{ 2 }=9{ q }^{ 2 }+6q+1$
    $=>{ n }^{ 2 }={ 3q\left( 3q+2 \right)  }+1$
    $=>{ n }^{ 2 }=3m+1 ,$ where $  m=q\left( 3q+2 \right) $
Now, if $n=3q+2$
    $=>{ n }^{ 2 }={ \left( 3q+2 \right)  }^{ 2 }$
    $=>{ n }^{ 2 }=9{ q }^{ 2 }+12q+4$
    $=>{ n }^{ 2 }={ 3q\left( 3q+4 \right)  }+4$
    $=>{ n }^{ 2 }={ 3q\left( 3q+4 \right)  }+3+1$
    $=>{ n }^{ 2 }=3m+1$ where $m=\left( 3{ q }^{ 2 }+4q+1 \right) $
Hence, ${ n }^{ 2 }$ integer is of the form $3m$ and $3m+1$ not $3m+2$

Multiple choice maths real number real numbers on number line fundamental theorem of arithmetic common factors and hcf

A rectangular veranda is of dimension $18$m $72$cm $\times 13$ m $20$ cm. Square tiles of the same dimensions are used to cover it. Find the least number of such tiles.

  1. $4290$
  2. $4540$
  3. $4620$
  4. $4230$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The edge of rectangular veranda are $18\ m\ 72\ cm=1872\ cm$ and $13\ m\ 20\ cm=1320\ cm$.


On taking $HCF$ of $1872$ and $1320$, we get

$HCF=24$

Therefore,
No. of tiles required $=$ $\dfrac{Area\ of\ Veranda}{Area\ of\ tiles}$

                                  $=\dfrac{1872\times 1320}{24\times 24}$

                                  $=4290$

Hence, this is the answer.

Multiple choice maths real number real numbers on number line fundamental theorem of arithmetic common factors and hcf

Determine the HCF of $a^2 - 25, a^2 -2a -35$ and $a^2+12a+35$

  1. (a-5)(a+7)

  2. (a+5)(a-7)

  3. (a-7)

  4. (a+5)
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Since, $a^2 - 25 = (a-5)(a+5) $
$ a^2 -2a -35 = a^2 -7a +5a -35 $
                         $= a(a-7)+5(a-7) $
                         $= (a+5)(a-7) $
and
$a^2+ 12a + 35 =a^2 +7a +5a +35 $
                          $=(a+7)(a+5) $
Clearly HCF of $a^2 - 25, a^2 -2a -35$ and $a^2+ 12a + 35$ i.e $ (a-5)(a+5), (a+5)(a-7)$ and $(a+7)(a+5)$ is $a+5$
Option D is correct.

Multiple choice maths real number real numbers on number line fundamental theorem of arithmetic common factors and hcf

The LCM of 54 90 and a third number is 1890 and their HCF is 18 The third number is

  1. 36

  2. 180

  3. 126

  4. 108
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given the LCM two numbers 54, 90 and third number is 1890 and HCF is 18

Let the number is 18x because one factor is also 18 the common factor HCF
Then factor 54,90 ,18 =$18\times 3,18\times 5,18\times 18\times x$

$\therefore 18\times 3\times 5\times x=1890\Rightarrow 270x=1890\Rightarrow x=7$
Then third number is $18\times 7=126$